The Power of Hypothesis - Although a powerful statistical method, hypothesis testing can lead to false conclusions if applied incorrectly. - BioPharm International

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The Power of Hypothesis
Although a powerful statistical method, hypothesis testing can lead to false conclusions if applied incorrectly.


BioPharm International
Volume 21, Issue 6

TWO-SAMPLE T-TEST

The two-sample t-test is used to compare two different sample means. Similar to the one-sample t-test, the hypothesis can be either a one-sided or two-sided test. The t-distribution is required here because instead of population mean, two samples are being compared. For the two-sample t-test, the degrees of freedom are the number of observations for sample 1 (n1) plus the number of observations for sample 2 (n2) minus two. The formula for the two-sample t-test is shown in the following equation:




in which mean1 and mean2 are the two means being compared, s1 and s2 are the standard deviation of each mean, and n1 and n2 are their respective sample sizes.


Table 2. An example of a one-sided two- sample t-test for purity. The hypothesis is that the new lot is not less pure than the old lot (New ≥ Old).
If the value of t* is greater than the tabled value from the t-distribution, the two sample means are statistically different. An example of a two-sample t-test would be comparing the existing lot to a previous lot of material. Table 2 shows an example of a one-sided two sample t-test for purity. The hypothesis is that the new lot is not less pure than the old lot (New ≥ Old). The mean of the new lot is statistically less pure than the mean of the old lot (p = 0.003). The t* of 3.49 exceeds the tabled value for a 95% confidence level with 10 degrees of freedom of 1.812. Notice that the tabled value of a one-sided 95% confidence level is the same as the two-sided 90% confidence level.

Z-TEST

The z-test is probably the most misused and misunderstood statistical test in biopharmaceutical organizations. The misconception is that if the sample size is large enough (n > 30) then the z-test is appropriate. The z-test is restricted for only those situations where the population variance is known. Because it is practically impossible to know the population variance and computers can calculate the t-distribution for any sample size, it is preferred to use the t-test. If it is truly necessary to use the z-test, the formula is:




in which X-mean is the sample mean, μ is the theoretical population mean, σ is the population standard deviation, and n is the sample size used to estimate the mean.


Table 3. A comparison of the z-values to the t-values for sample sizes of 30, 60, and 120.
Notice that the formula for the z-test is similar to the formula for the one-sample t-test. The only difference between the two equations is the estimation of the standard deviation (square root of the variance). The t-test uses the sample standard deviation and the z-test uses the population standard deviation. The normal distribution values do not require a sample size because the standard deviation is known and, therefore, the tabled value is the same regardless of sample size. Table 3 shows a comparison of the z-values to the t-values for sample sizes of 30, 60, and 120. The t-distribution, though close to the normal distribution, does not converge to normal distribution until the sample size exceeds 120.


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